De Moivre’s Theorem (AQA A-Level Further Maths): Flashcards

$r^n(\cos(n\theta) + i\sin(n\theta))$

All integers (positive, negative, or zero)

Raised to power $n$: $r^n$

Multiplied by $n$: $n\theta$

$e^{i\theta} = \cos\theta + i\sin\theta$

$2\cos\theta = e^{i\theta} + e^{-i\theta}$

$2i\sin\theta = e^{i\theta} - e^{-i\theta}$

$\cos(-\theta) + i\sin(-\theta)$

Equate real and imaginary parts separately

$S = \frac{a(1-r^n)}{1-r}$